↑

Properties

Label 1-356-356.47-r1-0-0
Degree $1$
Conductor $356$
Sign $0.279 + 0.960i$
Analytic cond. $38.2575$
Root an. cond. $38.2575$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.755 + 0.654i)3-s + (0.959 − 0.281i)5-s + (0.281 + 0.959i)7-s + (0.142 + 0.989i)9-s + (0.959 + 0.281i)11-s + (0.755 + 0.654i)13-s + (0.909 + 0.415i)15-s + (−0.415 − 0.909i)17-s + (0.989 − 0.142i)19-s + (−0.415 + 0.909i)21-s + (−0.989 + 0.142i)23-s + (0.841 − 0.540i)25-s + (−0.540 + 0.841i)27-s + (0.281 + 0.959i)29-s + (−0.989 − 0.142i)31-s + ⋯
L(s)  = 1  + (0.755 + 0.654i)3-s + (0.959 − 0.281i)5-s + (0.281 + 0.959i)7-s + (0.142 + 0.989i)9-s + (0.959 + 0.281i)11-s + (0.755 + 0.654i)13-s + (0.909 + 0.415i)15-s + (−0.415 − 0.909i)17-s + (0.989 − 0.142i)19-s + (−0.415 + 0.909i)21-s + (−0.989 + 0.142i)23-s + (0.841 − 0.540i)25-s + (−0.540 + 0.841i)27-s + (0.281 + 0.959i)29-s + (−0.989 − 0.142i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 356 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 356 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(356\)    =    \(2^{2} \cdot 89\)
Sign: $0.279 + 0.960i$
Analytic conductor: \(38.2575\)
Root analytic conductor: \(38.2575\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{356} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 356,\ (1:\ ),\ 0.279 + 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.893233687 + 2.170771118i\)
\(L(\frac12)\) \(\approx\) \(2.893233687 + 2.170771118i\)
\(L(1)\) \(\approx\) \(1.724167119 + 0.6380551814i\)
\(L(1)\) \(\approx\) \(1.724167119 + 0.6380551814i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (0.755 + 0.654i)T \)
5 \( 1 + (0.959 - 0.281i)T \)
7 \( 1 + (0.281 + 0.959i)T \)
11 \( 1 + (0.959 + 0.281i)T \)
13 \( 1 + (0.755 + 0.654i)T \)
17 \( 1 + (-0.415 - 0.909i)T \)
19 \( 1 + (0.989 - 0.142i)T \)
23 \( 1 + (-0.989 + 0.142i)T \)
29 \( 1 + (0.281 + 0.959i)T \)
31 \( 1 + (-0.989 - 0.142i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.755 - 0.654i)T \)
43 \( 1 + (-0.281 + 0.959i)T \)
47 \( 1 + (-0.654 - 0.755i)T \)
53 \( 1 + (0.654 - 0.755i)T \)
59 \( 1 + (0.755 - 0.654i)T \)
61 \( 1 + (-0.540 + 0.841i)T \)
67 \( 1 + (0.654 - 0.755i)T \)
71 \( 1 + (-0.959 - 0.281i)T \)
73 \( 1 + (-0.142 + 0.989i)T \)
79 \( 1 + (-0.142 + 0.989i)T \)
83 \( 1 + (0.909 - 0.415i)T \)
97 \( 1 + (-0.959 + 0.281i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.51061217815014435857039562864, −23.691987895765359474598223920574, −22.61183810031204138102681979917, −21.693959837974827973179320433396, −20.590723090384930025577334042629, −20.09956393873388241976942311130, −19.10883754922778905343831749163, −18.01635583737989913671722924237, −17.55922421411810933793301338195, −16.49699321877576306277142663673, −15.110263331371714697362375973354, −14.20545127605826182833308077727, −13.67277522956009756520258174268, −12.94084886027403422509321609963, −11.66283493073188677114734635380, −10.51896172229585511715837939299, −9.6344111332309591543960822226, −8.58800261821759206391761822506, −7.664043575478609180790204364828, −6.582231768894205879046924261647, −5.87060522940767674835079989699, −4.105186133154967699055974168400, −3.19123330815339969640750221813, −1.79854202101924930887878277334, −1.00622622755388869476991931213, 1.55357251322821394064593216397, 2.404390721900827895494649097469, 3.70058851904824363848627006151, 4.880872627226720120682560869, 5.758606925433634534699929144579, 7.01446621640326440156507237979, 8.46446651060184656411176263103, 9.22823373183816927366920153717, 9.62774373030395360159350233246, 11.01236652472720377536152516586, 11.98263947279220122749611105829, 13.220577989985830873611008951349, 14.143568983548516866030690260930, 14.61809159037755365714446876475, 15.950441229792778180828780295145, 16.392302694758241497854439317273, 17.82961498281943779163683167159, 18.37423162378370648708631246583, 19.67862053168828224627395132845, 20.41108900345724968031979756645, 21.2867270049972783071248921994, 21.89699028046949906813902365483, 22.60608621675358497598541496401, 24.29194248748971974895338036142, 24.83194742111935450938671350487

Graph of the $Z$-function along the critical line